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60 111 METRIC SPACES

(a) and (b) imply (c)₅ for each one-point set {x} is open, hence the family
of sets {x} is an open covering of E, and a finite subfamily can only be a
covering of E if E is finite. On the other hand, (c) implies both (a) and (b),
for each one point set being closed, every subset of E is closed as finite union
of closed sets, hence every subset of E is open, and therefore E is homeo-
morphic to a discrete space. Finally, as there is only a finite number of open
sets, E is compact.

(3.16.4) In a compact metric space E, any infinite sequence (x_n) which has only
one cluster value a converges to a.

Suppose a is not the limit of (x_n); then there would exist a number a > 0
such that there would be an infinite subsequence (x_nt) of (x_n) whose points
belong to E — E(a; a). By assumption, this subsequence has a cluster value
4, and as E - E(a; a) is closed, b belongs to E - B(a; a) by (3.13.7). The
sequence (x_n) would thus have two distinct cluster values, contrary to
assumption.

(3.16.5) Any continuous mapping f of a compact metric space E into a metric
space E' is uniformly continuous.

Suppose the contrary; there would then be a number a > 0 and two
sequences (x_n) and (y_n) of points of E such that d(x_n,_>y_n) < l/n and
d'(f(x_n),f(y_n)) ^ ^a- We.can find a subsequence (x_nk) converging to a point a,
and as d(x_nk,y_nk)< l/n_k, it follows from the triangle inequality that the
sequence (y_nk) also converges to a. But/is continuous at the point a, hence
there is a 5 > 0 such that d'(f(a),f(x)) < a/2 for d(a, x) < §. Take k such
that d(a,x_ni)<6, d(a,y_nk)<S; then d'(f(x_Hk),f(y_nk)) < a contrary to the
definition of the sequences (x_n) and (y_n).

(3.16.6) Let E be a compact metric space, (U_A)_A6L an open covering ofE.
There exists a number a > 0 such that any open ball of radius a is contained in
at least one of the U^ ("Lebesgue's property").

For every x 6 E, there exists an open ball B(.x; rj contained in one of the
sets U_A. As the balls #(x; r_x/2) form an open covering of E₉ there exist a
finite number of points *, e E such that the balls Bfo; r_xJ2) form a covering
of E. If a > 0 is the smallest of the numbers r_xJ2₉ it satisfies the required
property: indeed, every x e E belongs to a ball Bfo; r_X|/2) for some /, hence
B(x; a) is contained in Bfo; r_xt) since a < rJ2; but by construction B(x,; r_x)
is contained in some U_A.



	60 111 METRIC SPACES (a) and (b) imply (c)₅ for each one-point set {x} is open, hence the family of sets {x} is an open covering of E, and a finite subfamily can only be a covering of E if E is finite. On the other hand, (c) implies both (a) and (b), for each one point set being closed, every subset of E is closed as finite union of closed sets, hence every subset of E is open, and therefore E is homeo- morphic to a discrete space. Finally, as there is only a finite number of open sets, E is compact. (3.16.4) In a compact metric space E, any infinite sequence (x_n) which has only one cluster value a converges to a. Suppose a is not the limit of (x_n); then there would exist a number a > 0 such that there would be an infinite subsequence (x_nt) of (x_n) whose points belong to E — E(a; a). By assumption, this subsequence has a cluster value 4, and as E - E(a; a) is closed, b belongs to E - B(a; a) by (3.13.7). The sequence (x_n) would thus have two distinct cluster values, contrary to assumption. (3.16.5) Any continuous mapping f of a compact metric space E into a metric space E' is uniformly continuous. Suppose the contrary; there would then be a number a > 0 and two sequences (x_n) and (y_n) of points of E such that d(x_n,_>y_n) < l/n and d'(f(x_n),f(y_n)) ^ ^a- We.can find a subsequence (x_nk) converging to a point a, and as d(x_nk,y_nk)< l/n_k, it follows from the triangle inequality that the sequence (y_nk) also converges to a. But/is continuous at the point a, hence there is a 5 > 0 such that d'(f(a),f(x)) < a/2 for d(a, x) < §. Take k such that d(a,x_ni)<6, d(a,y_nk)<S; then d'(f(x_Hk),f(y_nk)) < a contrary to the definition of the sequences (x_n) and (y_n). (3.16.6) Let E be a compact metric space, (U_A)_A6L an open covering ofE. There exists a number a > 0 such that any open ball of radius a is contained in at least one of the U^ ("Lebesgue's property"). For every x 6 E, there exists an open ball B(.x; rj contained in one of the sets U_A. As the balls #(x; r_x/2) form an open covering of E₉ there exist a finite number of points , e E such that the balls Bfo; r_xJ2)* form a covering of E. If a > 0 is the smallest of the numbers r_xJ2₉ it satisfies the required property: indeed, every x e E belongs to a ball Bfo; r_X\|/2) for some /, hence B(x; a) is contained in Bfo; r_xt) since a < rJ2; but by construction B(x,; r_x) is contained in some U_A.